[Paper Review] Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes

Leonid Petrov|arXiv (Cornell University)|Feb 17, 2012

Random Matrices and Applications40 references8 citations

TL;DR

This paper establishes a double contour integral formula for the correlation kernel of uniformly random Gelfand-Tsetlin schemes, enabling asymptotic analysis of random lozenge tilings on general polygons with arbitrary numbers of sides. It proves that local bulk behavior converges to ergodic translation-invariant Gibbs measures matching Kenyon-Okounkov limit shapes, and edge behavior converges to the Airy process, extending prior results beyond the hexagon case.

ABSTRACT

A Gelfand-Tsetlin scheme of depth N is a triangular array with m integers at level m, m=1,...,N, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed N-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov in arXiv:math-ph/0507007. We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

Motivation & Objective

• To develop a general framework for asymptotic analysis of random lozenge tilings beyond the hexagon case.
• To establish a double contour integral expression for the correlation kernel of uniformly random Gelfand-Tsetlin schemes of arbitrary depth N.
• To analyze the local bulk and edge asymptotics of random tilings on polygons with arbitrarily many sides.
• To confirm that the limiting local behavior matches ergodic translation-invariant Gibbs measures with slopes corresponding to tangent planes of Kenyon-Okounkov limit shapes.
• To prove that edge behavior is governed by the Airy process, extending previous results limited to the hexagon or one-dimensional ensembles.

Proposed method

• Derive a double contour integral formula for the determinantal correlation kernel of uniformly random Gelfand-Tsetlin schemes with fixed Nth row.
• Use the kernel to perform asymptotic analysis in the bulk and at the edge of the limit shape under global scaling.
• Apply contour deformation techniques and critical point analysis to handle different cases based on the location of the double critical point w_c.
• Employ particle-hole involution and scaling limits to handle cases where Ω(w_c) < 0 or Ω(w_c) > 1.
• Use steepest descent methods and residue analysis to extract limiting behavior in the bulk and edge regimes.
• Relate the scaling limits to known universal processes: ergodic Gibbs measures in the bulk and the Airy process at the edge.

Experimental results

Research questions

• RQ1 Can a universal kernel formula be derived for random Gelfand-Tsetlin schemes that enables asymptotic analysis beyond the hexagon case?
• RQ2 Do the local bulk statistics of random lozenge tilings on general polygons converge to ergodic translation-invariant Gibbs measures matching the tangent planes of the limit shape?
• RQ3 Is the edge behavior of random tilings governed by the Airy process for polygons with arbitrary numbers of sides?
• RQ4 Can the asymptotic behavior at cusps of the limit shape be described by the Pearcey process?
• RQ5 How does the absence of orthogonal polynomial structure in general polygons affect the asymptotic analysis, and can this be overcome?

Key findings

• A double contour integral formula is derived for the correlation kernel of uniformly random Gelfand-Tsetlin schemes of depth N with arbitrary fixed Nth row.
• The local bulk asymptotics of random lozenge tilings converge to ergodic translation-invariant Gibbs measures whose slopes match the tangent planes of the Kenyon-Okounkov limit shape.
• At the edge of the limit shape, the asymptotic behavior converges to the Airy process, confirming universality in the edge regime.
• The method overcomes the lack of orthogonal polynomial structure in general polygons by introducing a new kernel formula and contour deformation techniques.
• The convergence to the Airy process is established in the full space-time sense, not just in static configurations, extending prior results limited to one-dimensional ensembles.
• The analysis covers polygons with arbitrarily many sides, allowing limit shapes and frozen boundaries of arbitrary algebraic degree.